Thursday, October 17, 2013

Kant on our knowledge of arithmetic and geometry

To: Scott B.
From: Geoffrey Klempner
Subject: Kant on our knowledge of arithmetic and geometry
Date: 4th January 2011 12:14

Dear Scott,

Thank you for your email of 22 December, with your essay for the University of London BA Modern Philosophy: Spinoza, Leibniz, Kant module, in response to the question, 'Explain and assess Kant's arguments for thinking that our knowledge of arithmetic and geometry is neither analytic nor a posteriori.

You have offered an account of Kant's general reasons in the Critique of Pure Reason for seeking to establish the existence of synthetic a priori truths, with particular reference to his claim that the truths of arithmetic and geometry are synthetic a priori, summarizing his views about the nature of time and space as 'a priori intuitions'.

What you haven't done is give any assessment of Kant's claims regarding arithmetic and geometry. However, this is a very difficult topic. I can imagine someone who has not studied Kant finding your essay useful and instructive.

But let's concentrate on the question of assessment. Is Kant right? You offer no opinion at all on this question.

Let's start by looking at some evidence. It was not until the nineteenth century that relative consistency proofs of non-Euclidean geometries were put forward. (Proofs to the effect that if Euclidean geometry is consistent then non-Euclidean geometry is consistent.) This appears directly to contradict Kant's claim that we know the truth of (Euclidean) geometry a priori. Modern physics and cosmology assume the truth of non-Euclidean geometry (the so called 'curvature' of space, which exhibits the properties of Reimanian geometry). We can't criticize Kant for his lack of knowledge. In his time, there was no evidence that cast doubt on Newtonian mechanics. Now, we know different. But shouldn't he have been able to *anticipate the possibility* that Euclidean geometry might not describe correctly the properties of physical space?

One argument which can be made in defence of Kant would be to say that, even if it is true that non-Euclidean geometry describes the properties of physical space, our 'a priori forms of intuition' require us to perceive space as Euclidean. In other words, we are incapable, owing to the constitution of our minds, of grasping the nature of space as it is in reality. However, for Kant, that would suggest that modern physics and cosmology are sciences of a noumenal world of 'things in themselves' of which we can have no knowledge. The only alternative would be to add a third 'world' in between Kant's phenomenal and noumenal worlds. Euclidean space is phenomenally real; non-Euclidean space is physically real; while the noumenal world lacks spatial properties entirely. That sounds a desperate fix.

Another attack on Kant's view of space -- this time, his claim about the a priori necessary unity of space -- was made by Anthony Quinton in his well-known paper, 'Spaces and Times' (look this up in Google). Quinton imagined a scenario where I fall asleep and wake up in an idyllic fishing village, where I have a completely different life. When I go to sleep there, I wake up in Sheffield. Quinton argued that my ability to describe my experience coherently does not require any assumption about the spatial relation between Sheffield and the idyllic fishing village. They could be on different planets, or even in different universes for all I can ever discover.

There are arguments that can be put forward against Quinton: the fact that my description of my 'double life' seems to make sense does not prove its coherence, in any deep sense. At any given time, say, when I am in Sheffield, all the evidence can gather, including reports of witnesses, is that when I go to sleep, I remain asleep in my bed, although evidently enjoying very pleasant dreams. Exactly the same applies, of course, when I am in my fishing village. In other words, Quinton has not 'described a possible experience', but merely described something which *seems* possible, but which he has not proved to be coherent.

You say something at one point in relation to Kant's argument for the a priori truth of causality which is relevant here. 'For instance, every event must have a cause because an uncaused event could not be experienced.' Really? According to current physical theory we can experience an uncaused event -- e.g. a random click from a Geiger counter. You hear the click, there's no doubt there. There doesn't need to be a causal explanation of why the click occurred just then in order for you to experience it. In fact, the Transcendental Deduction, the Analogies of Experience and the Refutation of Idealism Kant builds a case for determinism, on the grounds that an account of experience in a world where determinism failed would have fatal gaps, it would be deeply incoherent. But, once again, that takes some arguing.

Strawson in 'The Bounds of Sense' argues that Kant's account of the nature of space and time, and his transcendental deduction of the concepts of substance and cause only succeed in establishing the necessity for *sufficient regularity* in the world of our experience. We must be able to reliably identify causes and effects, and trace the paths of spatio-temporal particulars, identifying them as 'the same again' on different occasions. But this does not require the truth of determinism.

What we can say in Kant's favour, is that it is a priori true, as a necessary condition for the possibility of constructing a world on the basis of experience, that we must assume some form of space, which has some geometry, not necessarily three-dimensional Euclidean geometry. That's still an important result. Kant is right to reject the idea that we can 'abstract' space from our experiences, because there would be no experience at all if we were not able to regiment our data according to some objective conceptual schema. However, he has overstated the case in arguing for the synthetic a priori truth of geometry.

I haven't discussed the case of arithmetic. This is another lacuna in your essay. Here, the focus of argument is on Kant's belief in the connection between numbers and the 'a priori form of inner sense', i.e. time. In his 'Foundations of Arithmetic' Frege famously mounted a strong attack on Kant's view of arithmetic. Although Russell and Frege's 'logicist' program (the reduction of arithmetic to logic) failed to fulfil its promise, there are not many today who would accept Kant's view of arithmetic.

All the best,

Geoffrey